Classification of undulated wavefront aberration in projection optics by considering its physical effects

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1 46 5, May 2007 Classification of undulated wavefront aberration in projection optics by considering its physical effects Masato Shibuya, MEMBER SPIE Nobuaki Watanabe Masayuki Yamamoto Toshihumi Fukui Hiromi Ezaki Tomohiro Kiire Suezou Nakadate Tokyo Polytechnic University Faculty of Engineering 1583, Iiyama, Atsugi-shi Kanagawa, , Japan Abstract. Although wavefront aberration in stepper projection optics has been classified with respect to its spatial frequency on the pupil, the physical meaning of this classification has not been clarified. In this paper, we show that wavefront aberration can be classified into figure aberration, random aberration, and nonconserved aberration, by taking into consideration the theoretical effect of undulated wavefront aberration with respect to its spatial frequency. We also show that the predictions of this theoretical classification coincide with both the results of numerical simulations for random aberration and the experimentally measured values of local flare size. Since our classification has a clear physical meaning, it will be valuable and applicable in developing not only stepper projection optics but also other more general optics Society of Photo- Optical Instrumentation Engineers. DOI: / Subject terms: flare; local flare; wavefront aberration; optics; lithography; random aberration; conserved aberration; isoplanatism. Paper R received May 12, 2006; revised manuscript received Oct. 31, 2006; accepted for publication Nov. 16, 2006; published online May 10, Introduction As the pattern feature size of integrated circuits ICs has become finer, the aberration in stepper projection optics has been actively discussed by many authors. Especially since it was discovered that the critical dimension CD in the local flare of the image pattern size is dependent on the neighboring aperture size, 1 3 there has been great interest in the high-frequency component of wavefront aberration. Therefore, in many discussions concerning stepper projection optics, wavefront aberration has been classified with respect to its spatial frequency on the pupil. 4 9 However, the physical meaning of this classification is not clear. Considering randomness and correlation function of wavefront variables, Noll discussed statistically the optical image quality for mid- and high-spatial-frequency components of wavefront aberration. 10 Harvey et al. independently dealt with the scattering effects on optical surface from a similar point of view. 11,12 Moreover, Youngworth and Stones discussed the effect of the mid-spatialfrequency terms on general optical imaging, by using the concept of random wavefront errors and perturbation methods to evaluate wavefront errors of total optics. 13 Their theory reveals that the conventional performance measures such as the modulation transfer function MTF can be approximately estimated from random wavefront errors in the mid-spatial-frequency region. We have also discussed the MTF especially in connection with the relation between the spatial frequency of the image and that of the wavefront aberration. 14 The concept of random aberration introduced by us is similar to that of the mid-frequency aberration of Youngworth and Stones but is a little different from that of /2007/$ SPIE Noll or Harvey in that they consider the correlation of wavefronts even in the mid-frequency region. In this paper, we give a fundamental discussion of wavefront aberration in the field of a stepper projection lens. We introduce new concepts and show that wavefront aberration can be classified into figure aberration, random aberration, and nonconserved aberration, taking into consideration the theoretical effect of wavefront aberration with respect to its special frequency. 15,16 Conserved aberration thus consists of figure aberration and random aberration. The boundaries among the aberration classes are approximately formulated. Although there have been already some discerning statements on the propagation of fine wavefront aberration, 17,18 to our knowledge this is the first time that conserved and nonconserved aberrations have been introduced explicitly and defined. In addition we show that the predictions based on this theoretical classification coincide with both the results of numerical simulations of random aberration and the experimentally measured values of local flare size. Since this classification exhibits a clear physical meaning, it will be valuable and applicable in developing not only stepper projection optics but also other, more general optics such as space telescopes and microscope objectives. Further, it will be useful in developing both interferometers for measuring wavefront aberration in optics and computer programs for optical simulations. 2 Theory 2.1 Figure Aberration and Random Aberration In a first-order approximation, we can understand optical imaging as an interference between zeroth-order diffracted light direct light and first-order diffracted light. The fundamental configuration in projection optics is shown in Fig. 1. When the reticle is illuminated from an incoherent

Fig. 1 In the case of fine undulated wave aberration, the phase difference between zeroth and first orders is randomly distributed. There are M undulations per unit of NA. Fig.

$source, zeroth- and first-order diffracted light go through the projection lens, and they interfere with each other on the wafer. There are many pairs of zeroth- and first-order light rays.$

2 Fig. 1 In the case of fine undulated wave aberration, the phase difference between zeroth and first orders is randomly distributed. There are M undulations per unit of NA. Fig. 2 The schematic configuration for projection optics and propagation of undulated wavefront aberration. source, zeroth- and first-order diffracted light go through the projection lens, and they interfere with each other on the wafer. There are many pairs of zeroth- and first-order light rays. When the undulation pitch of wavefront aberration becomes fine, the phase differences between zerothand first-order light can be regarded as being distributed randomly. In this case, the effect of wavefront aberration on image quality can be properly estimated only by using the root mean square rms of the wavefront aberration. On the other hand, when the undulation pitch is not fine, the image quality will depend on the figure in the wavefront aberration and cannot be predicted from the rms alone. Introducing the parameter M as the undulation number in the pupil radius in the projection lens and the parameter as the coherence factor, we assume that when 2M 10, 1 the phase difference can be regarded as being distributed randomly and the effect of this wavefront aberration can be determined by the rms. 3,15,16 Therefore, we call a wavefront aberration that satisfies Eq. 1 a random aberration. 3,15,16 On the other hand, when 2M 10, 2 the effect of this wavefront aberration will depend on its own figure. Therefore we call it figure aberration. 2.2 Nonconserved Aberration and Conserved Aberration Characterization As the pitch of wavefront aberration becomes finer, the figure representing the wavefront aberration itself will change continuously during its propagation in the optics. 17,18 For simplicity, let us consider the optics shown in Fig. 2. This is called f-f optics and is made up of a reticle object, front optics in a single lens, a stop, rear optics in a single lens, and an image on a wafer. The focal length of the front optics is f 1, and that of the rear optics is f 2. When fine undulation occurs on the surface in the front optics and its pitch is p, we can derive the condition in which the wavefront aberration propagates to the stop without changing its figure as follows: The wavefront aberration can be regarded as being constant over one undulation pitch, and so we estimate the diffraction effect caused by a small aperture equal in size to this undulation pitch. The diffracted light from this small aperture will spread over the stop as shown in Fig. 2. If the spread size is smaller than the aperture size itself, the wavefront figure will be conserved along with its propagation to the stop. Thus this conservation condition is represented by the following equation: P f 1 P. 3 When this equation is satisfied, the aberration is called conserved. 3,13 Conserved aberration is further divided into figure aberration and random aberration, which have been discussed in Sec On the other hand, when P f 1 P, 4 the figure of wavefront aberration is not conserved. Then we refer to it as nonconserved aberration. Even in the case of nonconserved aberration, since the power spectrum can be conserved along with its propagation, 14 the wavefront aberration does not flatten Conservation condition and isoplanatic condition As shown in Fig. 3, when wavefront aberration with a pitch of P occurs on the stop pupil, the point spread function Fig. 3 Local flare caused by undulation in wavefront aberration on the stop

3 Fig. 4 Local flare caused by undulation in wavefront aberration on the stop. will spread over the wafer and cause the phenomenon called flare. The flare radius L F can be roughly estimated with the equation L F P f 2. Many optical imaging theories implicitly or explicitly assume that the isoplanatic condition is fulfilled. This condition means that the point spread function is unchanged by movement of the image point within a region of size equal to the value of the point spread function itself. In Fig. 4, consider object points O and A. The distance between the image points O and A is equal to L F, and thus the distance between O and A is L F0 =L F f 1 / f 2. By comparing the optical path AA with OO, the isoplanatic condition that the wavefront aberration of the object point O coincides with that of the object point A can be expressed as f 1 f 1 P L F0 = L F f 2 P f 2 = f 2 P f 1. 6 This equation exactly coincides with Eq. 3 in the aberration conservation condition. In other words, when an aberration is conserved during its propagation in the optics, the flare can be regarded as the part of the isoplanatic point spread function. On the other hand, when it is not conserved, flare is not isoplanatic. Therefore the characteristic of the flare in a nonconserved aberration is qualitatively different from that in a conserved aberration. In the preceding discussion, we consider only cases in which fine undulation is caused on the front lens. In principle this discussion can be extended to other cases. However, it will be necessary to discuss those other cases more precisely, especially when an undulated surface is near the reticle or the wafer. This will be a project for future research. 3 Confirming the Validity of Random Aberration Conditions by Simulation In order to confirm the validity of Eq. 2 for the condition of random aberration, we simulated the coherence curve for concrete wavefront aberration models by using PROLITH as shown in Fig. 5. A coherence curve is defined as the critical dimension CD variation with respect to the pattern pitch and is also called a through pitch imaging curve. Here 5 Fig. 5 Coherence curve for NA=0.75, =0.2, =193. the CD is the pattern width. In this figure the conditions are as follows: NA=0.75, =0.2, =193 nm, the nominal pattern width is that of a 150-nm black line, the pattern pitch is varied from 300 to 1500 nm, and the standard threshold intensity is the intensity that creates an exact 150-nm width with a pattern pitch of 5000 nm. The wavefront aberration has a sinusoidal ring shape, and its peak to valley P-V value is /10. The number of pupil points for simulation is We calculated cases in which the numbers of rings, M, are 2.5, 6.25,12.5, 25, 37.5, but show curves only for M =12.5,25,37.5 in order to keep them distinct. Also the difference CD between the case of M =25 and that of M =37.5 is shown. From Eq. 2, the condition for random aberration is given as M 10 2 =25. 7 According to Fig. 5, since the coherence curve for M =25 approximately coincides with that for M = 37.5, the condition 2 is valid and the concept of random aberration itself is meaningful. There are many pairs of zeroth-order light and first-order light rays, as shown in Fig. 6. Since we are using ringshaped wavefront aberration, there is little correlation in the phase differences between zeroth- and first-order light. When becomes larger, this randomness assumption is not appropriate, especially when the pitch is large. However, the prediction from the condition of Eq. 2 is characteristic for the case of =0.6, as shown in Fig. 7. Because the random aberration condition predicted from Eq. 2 is M =10/2 =8.3, the coherence curve for M =12.5 roughly coincides with those for M =25 and M =37.5. In this figure we also show only the curves for M =6.25,12.5, and 25 in order to distinguish them clearly. The difference CD between the case of M =12.5 and that of M =25 is shown, too. 4 The Relation between Conserved Aberration and Observed Local Flare Size In Fig. 8, a fine pattern exists at the center of the 2L 0 aperture. The CD the pattern image width results from an aperture size 2L 0 but is independent of the aperture size when the 2L 0 is larger than the particular size 2L FC

Fig. 8 An actual pattern at the center of an aperture of size 2L 0 on the reticle. 1/4. Therefore, we can roughly estimate that f 1 =400 mm and f 2 =100 mm. Putting these values and =193 nm into Eq.

Therefore this phenomenon can be regarded as caused by the isoplanatic point spread function and is called the local flare. In other words, it is caused by conserved wavefront aberration. From Eq.

4 Fig. 8 An actual pattern at the center of an aperture of size 2L 0 on the reticle. 1/4. Therefore, we can roughly estimate that f 1 =400 mm and f 2 =100 mm. Putting these values and =193 nm into Eq. 9, L FC =70 m 10 is obtained. This value coincides well with the measured value. 3,6,8,9 Fig. 6 Ring-shaped sinusoidal wavefront aberration and the relation between zeroth- and first-order light. Therefore this phenomenon can be regarded as caused by the isoplanatic point spread function and is called the local flare. In other words, it is caused by conserved wavefront aberration. From Eq. 3 or Eq. 6, the finest pitch P C in conserved wavefront aberration at the stop is given by P C = f 1 1/2. 8 In this case, from Eqs. 5 and 8, the local flare size L FC can be expressed as L FC f 2 f 2 = P C f 1 1/2. 9 In typical stepper projection lenses, the distance from reticle to wafer is about 1000 mm and the reduction ratio is 5 Nonconserved Aberration and Measurement of the Wavefront Aberration in a Projection Lens When one measures the figure of a lens surface by using the Fizeau interferometer, in order to eliminate any diffraction effect on the lens surface itself, the detector and the lens surface should be conjugate to each other. If they are not conjugate, the interference pattern blurs, is not precise, and is sometimes disturbed near the edge of the aperture. When the wavefront aberration of optics is measured by using the configuration shown in Fig. 9, the ray goes through the stop twice, in the forward and backward paths. Since the stop in the forward path cannot be conjugate to that in the backward paths, it is impossible for the detector to conjugate with both these two stops simultaneously. Because the rms wavefront aberration is improved to be below 10 2, 19 the measurement precision is required to be 10 3 or better. Therefore it is very likely that the wavefront aberration in the projection lens cannot be measured precisely. Fig. 7 Coherence curve for NA=0.75, =0.6, =193. Fig. 9 Fizeau interferometer for measuring the wavefront aberration in projection optics

5 Table 1 Typical numerical boundaries for aberration classification. Conserved aberration: Figure aberration 2M=10/ =50 Random aberration D/ f 1 1/2 =200/ /2 =700 Nonconserved aberration However, in view of the characteristic of random aberration, the effect of random aberration can be estimated by the surface figure error and the inhomogeneity of the glass material. Therefore it is necessary to measure figure aberration but not random aberration. Moreover, since figure aberration is conserved aberration, it propagates from one stop to the other stop without changing its figure. Thus the Fizeau interferometer shown in Fig. 9 is useful for measuring wavefront aberration. By putting the typical values =193 nm, =0.2, f 1 =400 mm, and the stop diameter D=200 mm into Eq. 1 and Eq. 3, the numbers of undulations in the pupil diameter can be estimated for boundaries among the three kinds of aberrations and are listed in Table 1. Since the boundary between random aberration and nonconserved aberration is ten times higher than that between figure aberration and random aberration, figure aberration can be precisely measured using the configuration in Fig. 9. Moreover, in the region of nonconserved aberration, since its figure is not conserved along with its propagation, it is not meaningful to measure high-frequency aspects in the wavefront aberration of the total optical lens. Thus, referring to Table 1, measuring pupil points is approximately enough, and any larger number of points makes no improvement. This result is the key point in designing interferometers for measuring wavefront aberration and is also applicable when developing computer programs for optical simulations. Furthermore, the power spectrum of nonconserved aberration of the total optical lens can be estimated from those of its individual elements. 6 Summary By considering the physical properties in undulated wavefront aberration, we can theoretically classify wavefront aberration into figure aberration, random aberration, and nonconserved aberration. Conserved aberration consists of figure aberration and random aberration. Moreover, we have confirmed the validity of the theoretically predicted boundary between figure aberration and random aberration by simulating the coherence curves. Also, we have shown that the flare size estimated from the predicted boundary between random aberration and nonconserved aberration coincides with the measured local flare size. Therefore the concepts of figure aberration, random aberration, and nonconserved aberration are meaningful, correct, and useful. Because this proposed classification has clear physical meaning, it will be valuable in the production and development not only of projection lenses for IC manufacturing equipment but also of other optics such as space telescopes. We have also identified the most essential requirements in developing both the interferometer for measuring wavefront aberration in optics and computer programs for optical simulations. What still remains for future research is to discuss more precisely the case when the undulated surface is near the reticle or the wafer. References 1. J. P. Kirk, Scattered light in photolithographic lenses, Proc. SPIE 2197, C. Krautschik, M. Chandhok, G. Zhang, S. Lee, M. Goldstein, E. Panning, B. Rice, R. Bristol, and V. Singh, Implementing flare compensation for EUV masks through localized mask CD resizing, Proc. SPIE 5037, M. Shibuya, Effect of fine undulation of wavefront aberration, in Japanese, Jpn. J. Opt. 34 3, C. Progler and A. Wong, Zernike coefficients: are they really enough? Proc. SPIE 4000, K. Lai, C. Wu, and C. Progler, Scattered light: the increasing problem for 193 nm exposure tools and beyond, Proc. SPIE 4346, S. P. Renwick, S. D. Slonaker, and T. Ogata, Size-dependent flare and its effect on imaging, Proc. SPIE 5040, K. Matsumoto, T. Matsuyama, and S. Hirukawa, Analysis of imaging performance degradation, Proc. SPIE 5040, T. Kanda, Y. Shiode, and K. Shinoda, 0.85NA ArF exposure system and performance, Proc. SPIE 5040, T. Matsuyama, T. Ishiyama, and Y. Ohmura, Nikon projection lens update, Proc. SPIE 5377, R. J. Noll, Effect of mid- and high-spatial frequencies on optical performance, Opt. Eng. 18, J. E. Harvey and A. Kotha, Scattering effects from residual optical fabrication errors, Proc. SPIE 2576, J. E. Harvey and C. L. Vernold, Transfer function characterization of scattering surfaces: revisited, Proc. SPIE 3141, R. N. Youngworth and B. D. Stone, Simple estimates for the effects of mid-spatial-frequency surface errors on image quality, Appl. Opt. 39, M. Shibuya, K. Aoyagi, and S. Nakadate, Jpn. J. Opt. 32, in Japanese. 15. M. Shibuya, H. Ezaki, T. Fukui, N. Watanabe, and A. Nishikata, Random aberration and local flare, Proc. SPIE 5377, M. Shibuya, N. Watanabe, H. Ezaki, and S. Nakadate, Considering the flare by introducing the random aberration and non-conserved aberration, Proc. SPIE 5754, J. W. Goodman, Statistical Optics, Sec , John Wiley & Sons, New York J. W. Goodman, Introduction to Fourier Optics, Sec. 3.7, McGraw- Hill, San Francisco T. Matsuyama, I. Tanaka, T. Ozawa, K. Nomura, and T. Koyama, Improving lens performance through the most recent lens manufacturing process, Proc. SPIE 5040,

6 Masato Shibuya is a professor in the Department of Media and Image Technology, Faculty of Engineering, Tokyo Polytechnic University. He graduated from Tokyo Institute of Technology TIT in 1977 with a master s degree in physics and joined Nikon Corporation, where he designed space and lithography optics and studied resolution enhancement technology. He is an inventor of a phaseshifting mask. He received his PhD from the University of Tokyo in He joined Tokyo Polytechnic University in 2001, and his research interests include fundamental lens optics, optical lithography, and superresolution. Nobuaki Watanabe is an optical designer in the Optical Design Department, Core Technology Center, Nikon Corporation. He graduated from Tokyo Polytechnic University in 2006 with a master s degree in Engineering. His major interest is optical design for various fields. Hiromi Ezaki is a professor in the Faculty of Engineering, Tokyo Polytechnic University. She graduated from Ochanomizu Woman s University in 1986 with a Master Degree in Physics, and joined the University of Tokyo as an assistant professor. She received her PhD from Ochanomizu Woman s University in She joined Tokyo Polytechnic University in 1995, and her research interests are quantum optics, quantum information, and superresolution in optical lithography. Tomohiro Kiire is a graduate student in the doctoral program at Tokyo Polytechnic University. He graduated from Tokyo Polytechnic University in He was an optical designer at Mahk Corporation from 2001 to He graduated from Tokyo Polytechnic University in 2005 with a master s degree in engineering. He was an assistant professor at Tokyo Polytechnic University from 2005 to He reentered the graduate school there in His research interests are optical measurement and instrumentation. Masayuki Yamamoto is a control software engineer in the Equipment Control Department of Sigmameltec Ltd. He graduated from Tokyo Polytechnic University in His major interest is electric circuit and control software design. Toshihumi Fukui is a system engineer at Shishi Corporation. He graduated from Tokyo Polytechnic University in His major interest is computer programming for manufacturing equipment. Suezou Nakadate is a professor in the Department of Media and Image Technology, Faculty of Engineering, Tokyo Polytechnic University. He graduated from Tokyo Institute of Technology TIT in 1978 with a master s degree in engineering, and joined the optical instrumentation laboratory at the Institute of Physical and Chemical Research as a researcher. He received his PhD from TIT in He joined Tokyo Polytechnic University in 1989, and his research interests include optical measurement and instrumentation, digital image processing, and signal processing

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